Algebra Examples

$e^{- 0.002 t^{2}} (\sqrt{t} + t^{4})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\frac{d}{d t} [e^{- 0.002 t^{2}} (t^{\frac{1}{2}} + t^{4})]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = e^{- 0.002 t^{2}}$ and $g (t) = t^{\frac{1}{2}} + t^{4}$ .

$e^{- 0.002 t^{2}} \frac{d}{d t} [t^{\frac{1}{2}} + t^{4}] + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $t^{\frac{1}{2}} + t^{4}$ with respect to $t$ is $\frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [t^{4}]$ .

$e^{- 0.002 t^{2}} (\frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{2}$ .

$e^{- 0.002 t^{2}} (\frac{1}{2} t^{\frac{1}{2} - 1} + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e^{- 0.002 t^{2}} (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 5

Combine $- 1$ and $\frac{2}{2}$ .

$e^{- 0.002 t^{2}} (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 6

Combine the numerators over the common denominator.

$e^{- 0.002 t^{2}} (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $2$ .

$e^{- 0.002 t^{2}} (\frac{1}{2} t^{\frac{1 - 2}{2}} + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 7.2

Subtract $2$ from $1$ .

$e^{- 0.002 t^{2}} (\frac{1}{2} t^{\frac{- 1}{2}} + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 8

Differentiate using the Power Rule.

Tap for more steps...

Step 8.1

Move the negative in front of the fraction.

$e^{- 0.002 t^{2}} (\frac{1}{2} t^{- \frac{1}{2}} + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 8.2

Combine fractions.

Tap for more steps...

Step 8.2.1

Combine $\frac{1}{2}$ and $t^{- \frac{1}{2}}$ .

$e^{- 0.002 t^{2}} (\frac{t^{- \frac{1}{2}}}{2} + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 8.2.2

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$e^{- 0.002 t^{2}} (\frac{1}{2 t^{\frac{1}{2}}} + \frac{d}{d t} [t^{4}]) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 8.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

$e^{- 0.002 t^{2}} (\frac{1}{2 t^{\frac{1}{2}}} + 4 t^{3}) + (t^{\frac{1}{2}} + t^{4}) \frac{d}{d t} [e^{- 0.002 t^{2}}]$

Step 9

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = - 0.002 t^{2}$ .

Tap for more steps...

Step 9.1

To apply the Chain Rule, set $u$ as $- 0.002 t^{2}$ .

$e^{- 0.002 t^{2}} (\frac{1}{2 t^{\frac{1}{2}}} + 4 t^{3}) + (t^{\frac{1}{2}} + t^{4}) (\frac{d}{d u} [e^{u}] \frac{d}{d t} [- 0.002 t^{2}])$

Step 9.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{- 0.002 t^{2}} (\frac{1}{2 t^{\frac{1}{2}}} + 4 t^{3}) + (t^{\frac{1}{2}} + t^{4}) (e^{u} \frac{d}{d t} [- 0.002 t^{2}])$

Step 9.3

Replace all occurrences of $u$ with $- 0.002 t^{2}$ .

$e^{- 0.002 t^{2}} (\frac{1}{2 t^{\frac{1}{2}}} + 4 t^{3}) + (t^{\frac{1}{2}} + t^{4}) (e^{- 0.002 t^{2}} \frac{d}{d t} [- 0.002 t^{2}])$

Step 10

Differentiate.

Tap for more steps...

Step 10.1

Since $- 0.002$ is constant with respect to $t$ , the derivative of $- 0.002 t^{2}$ with respect to $t$ is $- 0.002 \frac{d}{d t} [t^{2}]$ .

$e^{- 0.002 t^{2}} (\frac{1}{2 t^{\frac{1}{2}}} + 4 t^{3}) + (t^{\frac{1}{2}} + t^{4}) e^{- 0.002 t^{2}} (- 0.002 \frac{d}{d t} [t^{2}])$

Step 10.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$e^{- 0.002 t^{2}} (\frac{1}{2 t^{\frac{1}{2}}} + 4 t^{3}) + (t^{\frac{1}{2}} + t^{4}) e^{- 0.002 t^{2}} (- 0.002 (2 t))$

Step 10.3

Multiply $2$ by $- 0.002$ .

$e^{- 0.002 t^{2}} (\frac{1}{2 t^{\frac{1}{2}}} + 4 t^{3}) + (t^{\frac{1}{2}} + t^{4}) e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Apply the distributive property.

$e^{- 0.002 t^{2}} \frac{1}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) + (t^{\frac{1}{2}} + t^{4}) e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11.2

Apply the distributive property.

$e^{- 0.002 t^{2}} \frac{1}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) + (t^{\frac{1}{2}} e^{- 0.002 t^{2}} + t^{4} e^{- 0.002 t^{2}}) (- 0.004 t)$

Step 11.3

Apply the distributive property.

$e^{- 0.002 t^{2}} \frac{1}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) + t^{\frac{1}{2}} e^{- 0.002 t^{2}} (- 0.004 t) + t^{4} e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11.4

Combine terms.

Tap for more steps...

Step 11.4.1

Combine $e^{- 0.002 t^{2}}$ and $\frac{1}{2 t^{\frac{1}{2}}}$ .

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) + t^{\frac{1}{2}} e^{- 0.002 t^{2}} (- 0.004 t) + t^{4} e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11.4.2

Raise $t$ to the power of $1$ .

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) + t^{1} t^{\frac{1}{2}} e^{- 0.002 t^{2}} \cdot - 0.004 + t^{4} e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) + t^{1 + \frac{1}{2}} e^{- 0.002 t^{2}} \cdot - 0.004 + t^{4} e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11.4.4

Write $1$ as a fraction with a common denominator.

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) + t^{\frac{2}{2} + \frac{1}{2}} e^{- 0.002 t^{2}} \cdot - 0.004 + t^{4} e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11.4.5

Combine the numerators over the common denominator.

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) + t^{\frac{2 + 1}{2}} e^{- 0.002 t^{2}} \cdot - 0.004 + t^{4} e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11.4.6

Add $2$ and $1$ .

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) + t^{\frac{3}{2}} e^{- 0.002 t^{2}} \cdot - 0.004 + t^{4} e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11.4.7

Move $- 0.004$ to the left of $t^{\frac{3}{2}} e^{- 0.002 t^{2}}$ .

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) - 0.004 \cdot (t^{\frac{3}{2}} e^{- 0.002 t^{2}}) + t^{4} e^{- 0.002 t^{2}} (- 0.004 t)$

Step 11.4.8

Raise $t$ to the power of $1$ .

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) - 0.004 t^{\frac{3}{2}} e^{- 0.002 t^{2}} + t^{1} t^{4} e^{- 0.002 t^{2}} \cdot - 0.004$

Step 11.4.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) - 0.004 t^{\frac{3}{2}} e^{- 0.002 t^{2}} + t^{1 + 4} e^{- 0.002 t^{2}} \cdot - 0.004$

Step 11.4.10

Add $1$ and $4$ .

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) - 0.004 t^{\frac{3}{2}} e^{- 0.002 t^{2}} + t^{5} e^{- 0.002 t^{2}} \cdot - 0.004$

Step 11.4.11

Move $- 0.004$ to the left of $t^{5} e^{- 0.002 t^{2}}$ .

$\frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}} + e^{- 0.002 t^{2}} (4 t^{3}) - 0.004 t^{\frac{3}{2}} e^{- 0.002 t^{2}} - 0.004 t^{5} e^{- 0.002 t^{2}}$

Step 11.5

Reorder terms.

$4 e^{- 0.002 t^{2}} t^{3} - 0.004 e^{- 0.002 t^{2}} t^{\frac{3}{2}} - 0.004 e^{- 0.002 t^{2}} t^{5} + \frac{e^{- 0.002 t^{2}}}{2 t^{\frac{1}{2}}}$